For imaging or communication purposes it is important to understand how waves propagate through a randomly scattering medium.
The quantities of interest can generally be expressed in terms of statistical averages.
Usually the first- and second-order moments of the Green's function are sufficient to characterize them.
However in some circumstances fourth-order moments are needed, for instance
for scintillation problems [1]}, [2]} or the analysis of intensity correlation-based imaging [3]}, [4]}.
For imaging with narrow or broad band signals it is also important to characterize multifrequency moments [5]}.
We consider here the
paraxial regime corresponding to high-frequency and long-range propagation of a wave beam.
The paraxial regime is physically relevant and it models many situations, for instance laser beam propagation [6]}, [7]}
or underwater acoustics [8]}.
The equations that govern the evolution of the fourth-order moments in the paraxial regime
have been known for a long time [9]}-[10]}.
The solution of the fourth-order moment problem
was recently analyzed and discussed in [2]}, [12]}
when the four Green's functions involved in the fourth-order moment are evaluated at the same frequency.
In this paper, we extend this result to the case when the four Green's functions have different frequencies.
This new result makes it possible to analyze a number of configurations in wave propagation and imaging.
Here we consider two main motivating applications:
